Group-delay-dispersive multilayer-mirror structures and method for designing same

ABSTRACT

Negative Group-delay-dispersion mirror (NGDD-mirror) multilayer structures include a generally-orderly arrangement of layers or groups of layers in which the function of certain individual layers or groups of layers can be recognized and defined. The structures are broadly definable as a rear group of layers which are primarily responsible for providing a desired reflection level and reflection bandwidth together with a low and smoothly varying reflection phase-dispersion, and a front group of layers which are primarily responsible for producing high reflection phase-dispersion necessary for providing a desired NGDD level and bandwidth. The front group of layers relies on multiple resonant trapping mechanisms such as two or more effective resonant-cavities, employing one or more sub-quarter-wave layers to shape the phase-dispersion for providing a substantially-constant NGDD. In certain embodiments of the structures, a base layer or substrate of a highly-reflective metal can be used to reduce the number of dielectric layers needed to provide high reflectivity.

TECHNICAL FIELD OF THE INVENTION

The present invention relates to generally to multilayer mirrors havingcontrolled phase-dispersion. It relates in particular to multipleresonant multilayer structures having phase-dispersion arranged toprovide a predetermined negative group-delay-dispersion for pulsedradiation incident thereon, particularly ultrafast laser pulses.

DISCUSSION OF BACKGROUND ART

Ultrafast lasers generate a series of short optical pulses. Temporalseparation of the pulses is determined by a round-trip time of lightcirculating in the resonant-cavity of the laser. If a highenergy-per-pulse or high pulse-separation time is required, it isdesirable to operate the laser with as long a resonant-cavity aspossible.

Unfortunately, in many applications of ultrafast lasers, such asincorporating the laser in a small instrument, a laser having a cavitylength of about 2 m or more is simply not practical. A practical lengthis about thirty centimeters (cm) or less. In certain applications, alength of 10 cm may be desirable. To “fold” a 2 m long cavity, usingmultiple reflections, in order to obtain a 10 cm longest physicaldimension would require more than twenty reflections, accordinglyminimizing reflection losses is important.

In most ultrafast lasers, a cavity loss in excess of 1.0% would lead tosignificant loss of output power. By way of example, in an ultrafastlaser having 10% outcoupling, a 1% cavity loss (per round-trip) equatesto about 10% loss of output power. Because of this, even if 99.9%reflecting fold-mirrors were used, more than about ten intra-cavityreflections therefrom per round-trip would produce significantoutput-power reduction.

Further, in order to support the ultrashort pulse length characteristicof an ultrafast laser, the laser must possess a total negativegroup-delay-dispersion, (negative GDD or NGDD) i.e., the sum of the GDDof the laser gain-medium and all cavity components must be negative. Ina simple arrangement of a laser cavity and dielectric material therein,such as, a gain medium and a mode locking device, total cavity GDD wouldbe positive, i.e., shorter wavelength light experiences a higherrefractive index and lower group velocity, and lags behind longerwavelength light. This causes lengthening of a laser pulse each roundtrip and prevents stable, short-pulse operation.

One means of avoiding this is to include one or more NGDD devices havingcollective negative GDD at least equal to, and preferably greater than,this positive GDD. Furthermore, if the laser is to be tunable over arange of wavelengths, the NGDD devices must be effective over that rangeof wavelengths.

Reflective NGDD devices which have been used with prior-art ultrafastlasers include Gires-Tournois Interferometer (GTI) mirrors. A GTI-mirroris a multilayer NGDD-mirror including a reflector, which comprises astack of alternating high and low refractive index dielectric layers,each layer generally having an optical thickness of one-quarterwavelength (one QWOT) at the nominal operating wavelength of the laser,and a single, thick, Fabry-Perot-like “spacer” layer (typically manywavelengths thick) of a dielectric material deposited on the reflector.A partially-reflecting multilayer stack may (optionally) be deposited onthe spacer layer. A GTI-mirror typically gives a constant negative GDDover only a relatively narrow wavelength range, for example, about fiftynanometers (nm). In a GTI-mirror, the NGDD is achieved by selectiveresonant trapping of certain wavelengths in the spacer layer. Such adevice is described extensively in a paper “Compression of Femto SecondOptical Pulses with Dielectric Multilayer Interferometers”, Kuhl et al.,IEEE Transactions in Quantum Electronics, QE-22, 1, pp 182-185, (January1986).

In U.S. Pat. No. 5,734,503 (Szipocs et al.) multilayer NGDD mirrorsdescribed as “chirped mirrors” are disclosed. One disclosed example ofsuch a mirror includes a substrate having a structure of more than 40layers deposited thereon. In this structure, essentially no two adjacentlayers have the same optical thickness. Two materials are used foradjacent layers, one having a relatively high refractive index and theother a relatively low refractive index. Throughout the structure, theoptical thickness between adjacent layers is substantial, with opticalthickness ratios up to about 2:1 not being uncommon. The thickness ofindividual layers is computer generated (optimized) from an initiallayer system described as “intuitive”. Increasing individual layers inthickness from the front to the back of the layer system, i.e., from theoutermost layer towards the substrate, or a fourier transform design issuggested, although no detail of such an initial layer system isdisclosed.

It is taught that, following optimization, apart from a trend ofincreasing optical thickness of a “reflective period” from the front tothe back of the layer system, the layer system does not have any orderlystructure. It is taught that nearly-constant NGDD is achieved withoutthe use of resonant trapping mechanisms in the structure, and resultssimply from different penetration depths of different wavelengths intothe structure. Such a mirror appears to be able to provide constant NGDDover a broader band of wavelengths than a GTI-mirror, for example, up toabout 150 nm, at a nominal center wavelength of about 800 nm, for a GDDof −45 fs².

While the Szipocs et al. NGDD-mirror appears to achieve a desired valueof nearly-constant NGDD over a bandwidth greater than has been achievedin devices of the GTI type which rely on a wide resonant-cavity toprovide NGDD, it would appear from consideration of optical multilayertheory that the mirror structure is far from that which would producethe highest possible reflectivity over the broadest bandwidth with thesame number of layers of the same materials.

It is well-known to designers of multilayer optical devices that thehighest reflectivity that can be obtained with a group of layers havingalternately high and low refractive index is achieved when all layers inthe group have the essentially the same optical thickness (an opticalthickness ratio of 1:1). Essentially, here meaning to the extent that isachievable considering refractive index dispersion in the materials.Departures from the 1:1 ratio will result in a lower reflectivity over anarrower bandwidth. The greater the departure the lower thereflectivity.

It would be advantageous to provide multilayer structures which achievedcomparable NGDD to the Szipocs et al. structures over the same orbroader bandwidth, while preserving sufficient order in the structuresthat the magnitude and bandwidth of reflectivity were not undulycompromised by any structural mechanisms or features necessary toprovide that NGDD. The present invention provides such structures andmethods for designing them.

SUMMARY OF THE INVENTION

The present invention is directed to providing a multilayer mirrorstructure for providing greater than a selected high reflectivity valueand substantially-constant or nearly-constant negativegroup-delay-dispersion over a selected band-of wavelengths. This isachieved, not by seeking to eliminate or avoid resonant trapping in thestructure, but to provide an arrangement of layers in the front of thestructure which intentionally causes selective resonant trapping ofcertain wavelengths within the selected band of wavelengths to occur.This selective resonant trapping occurs in two or more spaced apartsubgroups of layers in that arrangement.

In one aspect, the present invention comprises a substrate having amultilayer structure disposed thereon. The multilayer structure includesfirst and second pluralities of layers, the second plurality of layersbeing furthest from the substrate.

The first plurality of layers functions primarily to provide therequired high reflectivity value. The second plurality of layersfunctions primarily to provide a high reflection phase-dispersion forthe mirror within the selected band of wavelengths and, cooperative withthe first plurality of layers, to provide that the reflectionphase-dispersion of the mirror constantly increases over the selectedband of wavelengths from about the shortest to about the longestwavelength thereof, in a way which provides substantially constantnegative group-delay-dispersion across the band of wavelengths.

The reflective-phase-dispersion-controlling function of the secondplurality of layers is effected by arranging the layers such thatselective resonant trapping of certain wavelengths within the selectedband of wavelengths occurs in at least two longitudinally spaced-apartcavity groups of one or more layers within the second plurality oflayers. High phase-dispersion in the this context is understood to be aphase-shift on reflection difference between shortest and longestwavelengths in the selected band of wavelengths of least 180° (π).

In another aspect of the present invention, the first plurality oflayers can be arranged as essentially a quarter-wave stack ofalternating layers of two different transparent dielectric materials.The term essentially, here, means, for example, that there may be aterminal layer having less than one quarter-wavelength optical thicknessor that there is a minor variation, for example about ±10% of someaverage value, in the layers. By way of example, such a thicknessvariation can be somewhat random in nature and exist as a result ofcomputer optimization adjustments that have been found ineffectiveaccording to the optimization algorithm and not subsequently reset. Thevariation may be more orderly and progressive in layers adjacent thesecond plurality of layers. This may be done for forming an interfacebetween the first and second pluralities of layers.

Such a stack has a bandwidth about equal to or slightly less than abouta corresponding “normal” reflection bandwidth. A “normal” bandwidth,here, being defined as the bandwidth of a well-known quarter-wave stackof two different transparent dielectric materials wherein each of thelayers is about one QWOT at some so-called center-wavelength. As aquarter-wave stack provides a frequency-symmetrical spectral-response,the so-called center-wavelength is actually not centrally located in thereflection band in wavelength terms, but is that wavelength which is atthe frequency-midpoint of the reflection band.

In yet another aspect of the present invention, the first plurality oflayers may also be arranged as a broad-band mirror which, here, refersto a mirror having a reflection bandwidth broader than the corresponding“normal” bandwidth discussed above. An important characteristic of thebroad-band mirror is that it has a minimum reflection phase-dispersionover the selected bandwidth, preferably less than about 90° (π/2) andmore preferably about 60° (π/3) or even less. This, of course refers tothe phase-dispersion on a stand alone basis, i.e., without the secondplurality of layers, and seen in the intended direction of use of thecomplete inventive NGDD-mirror. An exactly-zero phase-dispersion is notnecessary, and, in the context or the present invention, may not even bepractical or possible. The form of what reflection phase-dispersion doesexist should be smoothly and monotonically varying, meaning that, withinthe selected band of wavelengths it should be free of ripples, abruptchanges of slope, changes of sign of the slope, discontinuities,singularities or the like.

The first plurality of layers defines the width over which highreflectivity in the inventive NGDD-mirrors can be obtained. As thehighly phase-dispersive multiple-resonant second plurality of layersprovides a high reflectivity only at shorter wavelengths in the selectedband of wavelengths, and, further, has a highest reflectivity which maybe an order of magnitude or more lower than that of the highestreflectivity value of the first plurality of layers, it merelysupplements the reflectivity in that band. The first plurality oflayers, arranged as discussed above, defines the maximum possiblebandwidth over which a particular value of nearly-constant NGDD can beprovided by the multiple-resonant second plurality of layers arrangedthereon. Short and long wavelength limits are imposed by abrupt changesin phase-dispersion preceding (at the short wavelength limit) andfollowing (at the long wavelength limit) the above-described smoothvariation of phase-dispersion at a relatively low level. Typically, thereflection phase-dispersion (slope) gradually decreases from the shortwavelength limit to an “effective center-wavelength” and then graduallyincreases from that “effective center-wavelength”. Considered anotherway, the reflection phase-dispersion of the first plurality must (seenfrom the direction of use, i.e., from the “front” of the mirror or theintended direction of incidence for radiation) be similar in form to thereflection phase-dispersion of a hypothetical “quarter-wave stack”having the same reflection bandwidth. Seen from the “rear” of themirror, the reflection phase-dispersion may have a complex formincluding rapid changes through one or more cycles of 360° (2 π).

In yet another aspect of the present invention, the second plurality oflayers is arranged with at least two spaced-apart groups orsub-structures of one or more layers. Each group provides, about thoselayers, the effect of a resonant-cavity for certain wavelengths in theselected band of wavelengths. The resonant-cavities can be formed, forexample, by an effect similar to the so-called index-conjugate orphase-conjugate effect, about a junction between (juxtaposition of) ahigh refractive index layer and a low refractive index layer, eachthereof having a similar thickness significantly less than aquarter-wavelength at a wavelength in the selected band of wavelengths.While such a sub-structure does not include a layer having an opticalthickness of one-half wavelength or more at a comparable wavelengthwhich could be defined as forming a “Fabry-Perot type” resonant-cavity,the resonant effect and accompanying amplification of electric-fieldintensity are essentially the same. This resonant effect is necessary toachieve the NGDD specification of the inventive GDD mirrors within thelimits imposed by the first plurality of layers on which it issuperposed.

The index-conjugate and other sub-quarter wavelength resonance-formingsub-structures of one or more layers are discussed in detail furtherbelow. One or more of the resonant sub-structures may even be aFabry-Perot type structure centered on a layer having an opticalthickness of about one-half wavelength in the reflection bandwidth ofinterest. Whatever the form of the resonant substructures, theireffective quality as resonators or “Q” is determined by the number oflayers between the structures and the refractive index ratio of thelayers.

It is emphasized here, that while resonant sub-structures arrangedaround layers less than one quarter-wavelength optical thickness arepreferred over half-wavelength or Fabry-Perot structures in NGDD-mirrorsin accordance with the present invention, this is because the morecomplex structure with thinner layers and multiple optical-boundaries isbelieved to provide a more flexible and smoother phase control. It isvery definitely not because of any desire to eliminate or avoidFabry-Perot type resonances as has been taught in the prior art to benecessary. As noted above, while the sub-quarter-wavelength-layer-basedresonant structures are physically different from Fabry-Perot typestructures, the resonant effect is similar to Fabry-Perot resonance andis necessary in NGDD-mirror structures in accordance with the presentinvention. Appropriate arrangement of the number and spacing of theseresonant substructures in the second plurality of layers defines themagnitude of the nearly-constant NGDD, and also establishes, withinlimits imposed by the form of the first plurality of layers, thebandwidth over which the nearly-constant NGDD is obtained.

A preferred method of a multilayer mirror structure design in accordancewith the present invention, the method comprises: entering into asuitably-programmed computer a starting structure including a firstportion thereof adjacent the substrate and arranged to provide the highreflectivity over at least a portion of the selected band ofwavelengths, and a second portion thereof superposed on the firstportion thereof and including at least two spaced-apart sub-structuresfor providing selective resonant trapping of wavelengths within theselected band of wavelengths; and then automatically optimizing, via thecomputer program, at least the thickness of layers of the startingstructure and thereby providing the multilayer mirror structure design.Some preferred examples of starting structures are discussed in detailhereinbelow.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, schematically illustrate a preferredembodiment of the present invention, and together with the generaldescription given above and the detailed description of the preferredembodiment given below, serve to explain the principles of theinvention.

FIG. 1 is a bar graph schematically illustrating a prior-art,chirped-mirror layer structure for providing a constant GDD of about −45fs² over a bandwidth between about 730 nm and 830 nm.

FIG. 2 is a graph schematically illustrating the computed reflection asa function of wavelength for the structure of FIG. 1.

FIG. 3 is a graph schematically illustrating the computed reflection-GDDas a function of wavelength for the structure of FIG. 1.

FIG. 4 is a bar graph schematically illustrating one example of anNGDD-mirror layer structure in accordance with the present invention forproviding a substantially constant NGDD of about −45 fs² and greaterthan 99.99% reflectivity over a bandwidth at least between about 730 nmand 870 nm.

FIG. 5 is a graph schematically illustrating the computed reflection asa function of wavelength for the NGDD-mirror structure of FIG. 4.

FIG. 6 is a graph schematically illustrating the computed reflection-GDDas a function of wavelength for the NGDD-mirror structure of FIG. 4.

FIG. 7 is a graph schematically illustrating the computed reflection andphase-shift on reflection as a function of wavelength for the 41-layerNGDD-mirror structure of FIG. 4.

FIG. 8 is a graph schematically illustrating the computed reflection andphase-shift on reflection as a function of wavelength for rear layers1-25 alone of the NGDD-mirror structure of FIG. 4.

FIG. 9 is a graph schematically illustrating the computed reflection andphase-shift on reflection as a function of wavelength for front layers26-41 alone of the NGDD-mirror structure of FIG. 18.

FIG. 10 is a graph schematically illustrating electric fielddistribution for radiation at 730, 790, and 870 nm wavelength in frontlayers twenty through forty of a prior-art, forty-layer, all-dielectric,all quarter-wave mirror arranged for peak reflectivity at 790 nm.

FIGS. 11A-C are graphs schematically illustrating electric fielddistribution for radiation at 730, 790, and 870 nm wavelengthrespectively in front layers twenty through forty-one of a theNGDD-mirror structure of FIG. 4.

FIG. 12 is a bar graph schematically illustrating a second example of anNGDD-mirror layer structure in accordance with the present invention forproviding a substantially constant NGDD of about −80 fs² and greaterthan 99.99% reflectivity over a bandwidth at least between about 740 nmand 840 nm.

FIG. 13 is a graph schematically illustrating the computedreflection-GDD as a function of wavelength for the NGDD-mirror structureof FIG. 9.

FIG. 14 is a bar graph schematically illustrating a third example of anNGDD-mirror layer structure in accordance with the present invention forproviding a substantially constant NGDD of about −80 fs² and greaterthan 99.99% reflectivity over a bandwidth at least between about 740 nmand 840 nm.

FIGS. 15A-C are graphs schematically illustrating electric fielddistribution for radiation at about 772, 825, and 845 nm wavelengthrespectively in front layers twenty-four through forty-eight of a theNGDD-mirror structure of FIG. 11.

FIG. 16 is a bar graph schematically illustrating a fourth example of anNGDD-mirror layer structure in accordance with the present inventionincluding a silver layer surmounted by 23 dielectric layers providing asubstantially constant NGDD of about −45 fs², and greater than 99.9%reflectivity over a bandwidth at least between about 730 nm and 870 nm.

FIG. 17 is a graph schematically illustrating reflectivity as a functionof wavelength for the structure of FIG. 16.

FIG. 18 is a bar graph schematically illustrating a simple, orderly,symmetrical eleven layer resonant structure with no layer having athickness greater than one-quarter wavelength at the resonantwavelength.

FIG. 19 schematically illustrating electric field distribution at theresonant wavelength in the structure of FIG. 18.

FIG. 20 is a graph schematically illustrating reflectivity as a functionof wavelength for an eleven layer structure (HL)² fH (LH)², with QWOT at790 nm wavelength, for values of f of 0.5, 0.4, 0.3, 0.2 and 0.1.

FIG. 21 is a bar graph schematically illustrating a sixth example of anNGDD-mirror layer structure in accordance with the present invention forproviding a substantially constant NGDD of about −50 fs² and greaterthan 99.90% reflectivity over a bandwidth at least between about 770 nmand 1000 nm.

FIG. 22 is a graph schematically illustrating the computed reflection asa function of wavelength for the NGDD-mirror structure of FIG. 21.

FIG. 23 is a graph schematically illustrating the computedreflection-GDD as a function of wavelength for the NGDD-mirror structureof FIG. 21.

FIG. 24 is a graph schematically illustrating the computed reflectionand phase-shift on reflection as a function of wavelength for a priorart 27 layer quarter-wave mirror.

FIGS. 25A and 25B are graphs illustrating electric field distributionfor radiation at about 950 nm and 1000 nm wavelength respectively infront layers twenty-six through forty-eight of a the NGDD-mirrorstructure of FIG. 21.

FIG. 26 is a graph schematically illustrating the computed reflectionand phase-shift on reflection as a function of wavelength for theNGDD-mirror structure of FIG. 21.

FIG. 27 is a graph schematically illustrating the computed reflectionand phase-shift on reflection as a function of wavelength for frontlayers 28-48 alone of the NGDD-mirror structure of FIG. 21.

FIG. 28 is a graph schematically illustrating the computed reflectionand phase-shift on reflection as a function of wavelength for rearlayers 1-27 alone of the NGDD-mirror structure of FIG. 21.

DETAILED DESCRIPTION OF THE INVENTION

Turning now to the drawings, FIG. 1 illustrates, in bar graph form, aNGDD-mirror structure according to above discussed teachings of Szipocset al. Each layer is shown as a vertical bar having a heightproportional to its optical thickness. A value of 1.0 is a quarter-waveoptical thickness or QWOT (at a wavelength of 790 nm in computations andresults discussed below). These optical thicknesses are taken from Table4 of the Szipocs et al. patent, that being an example of the disclosed“chirped-mirror”. The layers are assumed to have high and low refractiveindices of 2.31 and 1.45 respectively. There are 42 layers, thesubstrate being at the origin of the graph of FIG. 1, the first layer(on the substrate) being a low refractive index layer, layersalternating high and low refractive index thereafter. It was decided touse this prior-art structure as a comparison structure for an initialtest of an NGDD-mirror structure in accordance with the presentinvention as follows. The purpose of this test was to demonstrate theimportant and advantageous function of resonant mechanisms in an NGDDlayer structure

First, the reflection phase-angle (phase-shift on reflection) as afunction of wavelength for the structure of Table 1 was computed using acommercially available software, TFCalc™ from Software Spectral Inc., ofPortland, Oreg. Reflectance and reflection-GDD of the structure, asfunctions of wavelength, were computed using MACLEOD™ software availablefrom The Thin Film Center, of Tucson Ariz. The results are depicted inFIG. 2 and FIG. 3 respectively. Those skilled in the art will be awarethat the NGDD is the second differential with respect to frequency ofthe reflection phase-angle.

Next, an initial structure in accordance with the present invention wasloaded into the software. The structure was as follows.

Substrate (H/2 L H/2)¹⁵(L/2 H L/2)²(H/2 L H/2)²  (1)

where H and L represent one-quarter wavelength optical thickness or QWOTof respectively high and low refractive index material, by whichconvention, H/2 and L/2 represent one-eighth wavelength opticalthicknesses. Layers within parentheses are designated a group, and thesuperscript outside the parentheses is designated a group-repetitionnumber. This convention and minor variations thereof, (linguistic ordecimal, for example), are well known to those skilled in the art. Thebrief description above is offered for guidance in following structuralexamples of the present invention presented below.

Structure (1) is a 41 layer structure, adjacent eighth-wave layers ofthe same refractive index being consolidated to form a QWOT layer. H andL are assumed to have the same index values (2.31 and 1.45) of theprior-art structure of FIG. 1.

Adjacent layers (pairs) 31 and 32, and 36 and 37 (numbering from thesubstrate) are eighth-wavelength optical thickness layers of differentrefractive index. The structure causes resonance (selective resonanttrapping) for certain wavelengths in the regions of these eighth-wavelayer pairs. The resonance mechanism is based on what is known variouslyas a phase-conjugate or index-conjugate mechanism, the latter being amore accurate description. It is described in detail in U.S. Pat. No.3,528,726, where its use for making narrow bandpass-filters isdisclosed. Briefly, a structure:

(L/2 H L/2)^(n)(H/2 L H/2)^(n)  (2)

represented as a 2×2 matrix, is a unit matrix for some wavelength longerthan the QWOT (center) wavelength of the layers. Each group has adispersive, entirely-imaginary effective-refractive-index(effective-index) within some bandwidth defined by the high and lowrefractive index values of H and L. The effective-indices of each groupextend through the same range of magnitude but are opposite in sign andslope. A unit matrix is derived at a wavelength where theeffective-indices are conjugate. Resonance is a maximum at thisconjugate wavelength. A Fabry-Perot resonant-cavity is represented by aunit matrix at a wavelength at which it is an integer multiple ofhalf-wavelengths (twice that multiple of QWOTs) thick. We may describestructure (2) as forming a virtual-cavity, effective-cavity or“conjugate-cavity” to differentiate from the “real” (structurallyobvious) cavity of a Fabry-Perot etalon or GTI. It should be noted,here, that structure (2) is a particular case of a more generalstructure (aH bL aH)^(n)(aL bH aL)^(n), where 2a+b=2, which alsoprovides for resonance according to the same principle.

Continuing now with a discussion of initial structure (1), this initialstructure was computer-optimized to match the reflection phase-shift asa function of wavelength computed for the Szipocs et al. structure ofFIG. 1. Optimization targets were set in the range between 730 nm to 870nm. On the assumption that if this phase function were reproduced, sowould be the NGDD as a function of wavelength at lest in that wavelengthrange. The optimization method was the “variable metric” method. Itshould be noted in particular, that, all layers were allowed to bevariable in the optimization, and optimization was a single-parameter,phase-only optimization (reflection was not optimized).

The optimized structure is depicted graphically in FIG. 4 in the samemanner and units as the structure of FIG. 1. Odd-numbered layers, here,are high-refractive-index layers. It can be seen that layers 1-15 areessentially unchanged from their initial thickness and layers 16-27 arechanged only by a few (less than ten) percent. The “conjugate-cavity”structure as evidenced by layers 31 and 32 and 36 and 37 is stillclearly recognizable even with the layer thickness changes in layers28-41 resulting from the optimization.

Following the optimization of structure (1) to the reflectionphase-shift of the structure of FIG. 1, the reflectance andreflection-GDD, as functions of wavelength, of the optimized structurewere computed using the MACLEOD™ software. The results are depicted inFIG. 5 and FIG. 6 respectively.

Comparing the graphs of FIG. 3 with FIG. 6 it can be seen that, asexpected, the GDD values in the range of phase-optimization areessentially identical. Note that, outside this range, the prior-artstructure shows an extreme perturbation of the GDD in the region of 720nm. There is no such perturbation, however, in the GDD curve of theinventive structure.

Now comparing the graphs of FIG. 2 and FIG. 5 it can be seen that theorderly nature of the inventive structure provides that for essentiallythe same number of layers (actually one less) having the same refractiveindex values, the inventive structure provides a much higher reflectionover a wider bandwidth, than does the disorderly,widely-varying-layer-thickness, prior-art structure. For example, theprior-art structure reaches 99.99% reflectivity only at about 790 nmfalling to about 99.8% and 99.95% at about 730 nm and 870 nmrespectively. The GDD perturbation at 720 nm of the prior-art structureis associated with a severe dip in reflectivity in this region. Theinventive structure, however, has a reflectivity greater than 99.99%throughout this entire range and greater than 99.999% at peak.

The structure of FIG. 4 can be described as having a rear (mirror orreflector) portion thereof, for example layers 1-23, which primarilyprovides a desired reflectivity over a desired bandwidth; and a front(phase-retarder) portion thereof, for example layers 24-41, whichprimarily provides the phase-retardation, and form of thatphase-retardation necessary to result in a nearly-constant negative GDD.The term primarily, here, recognizes that the functions of the portionsare not absolutely exclusive. Clearly, one portion must complement theother. For, example, whatever phase-retardation dispersion is in themirror portion (in the structure shown this would typically berelatively flat) adds to that of the front portion. Clearly also, thephase-retarder layers have some effect on the reflectivity of the wholestructure, as wavelength-selective reflection is an essential part ofthe phase-retardation function. In this regard, those skilled in the artwill recognize that the division of the structure into mirror andphase-retarder portions is somewhat arbitrary. Certain layers, forexample, layers 24-27 may be considered interface layers or interfaceportion between the mirror and phase-retarder portions.

This structural and functional concept is illustrated with reference toFIG. 7, FIG. 8. and FIG. 9. FIG. 7 is a graph schematically illustratingthe computed reflection and (curve 7A) and phase-shift (curves 7B) onreflection as a function of wavelength for the 41-layer NGDD-mirrorstructure of FIG. 4. Here, it should be noted that the phase curves showa general, continuing increase in slope. This is required to provide anegative GDD. Also worthy of note are two sharp, narrow dips inreflection 7C and 7D in the wavelength range 650 to 700 nm, andcorresponding phase discontinuities 7E and 7F in curve 7B.

FIG. 8 is a graph schematically illustrating the computed reflection(curve 8A) and phase-shift on reflection (curves 8B) as a function ofwavelength for rear layers 1-25 alone of the NGDD-mirror structure ofFIG. 4. Those skilled in the art will recognize that the reflectioncharacteristic is essentially that of a “quarter-wave stack” dielectricmirror structure having a center wavelength at about 790 nm which is theQWOT wavelength for layers of the structure of FIG. 4. Here, it shouldbe noted that from an abrupt change in slope at a short wavelengthband-extremity 8S at about 700 nm, phase-shift slope first gentlydecreases to about 0°/360° at 790 nm and then gently increases towardanother abrupt change 8L in slope beyond 900 nm. Within the rangebetween about 700 and 900 nm phase change is only about 60° (π/3). Therapidly varying phase-shift and reflectivity in the wavelength range 650to 700 nm is that which gives rise to the reflectivity variations andphase discontinuities 7E and 7F in curve 7B.

FIG. 9 is a graph schematically illustrating the computed reflection(curve 9A) and phase-shift (curve 9B) on reflection as a function ofwavelength for front layers 26-41 alone of the NGDD-mirror structure ofFIG. 4. Here, it should be noted that the reflection, in general, is lowby comparison with the reflectivity of the reflector portion, and isslowly and smoothly varying, progressively reflecting shorterwavelengths in preference to longer wavelengths as might be expected.The phase-shift change across the wavelength region from about 700 toabout 900 is almost 360° (almost a full cycle of 2 π). The slope of thephase-shift change begins to decrease at wavelengths longer than about790 nm. This is compensated by (or rather is compensating for) theincreasing slope of the phase of the rear (mirror) portion to providethe composite phase-shift of the entire NGDD-mirror structure.

The importance of the resonance mechanism (resonant trapping mechanism)in NGDD-mirror structures in accordance with the present invention isnext discussed beginning with reference to FIG. 10. Here, thedistribution of electric field intensity (E²) for radiation at 730, 790,and 870 nm wavelength (curves 10A-C) in front layers twenty throughforty of a prior-art, forty-layer, all-dielectric, all quarter-wavemirror arranged for peak reflectivity at a wavelength of 790 nm, isschematically, graphically depicted. The well-known characteristicamplification of electric field in entrance-medium M in front of themirror is depicted. As there is no resonant mechanism (sub-structure) inthe layer structure for those wavelengths, E² within the structure atall wavelengths is progressively attenuated with depth in the structure.In the periodically-varying field, each peak has a substantially loweramplitude than (less than half of) the preceding peak, those peaks beingbarely visible deeper than layer number 30.

Referring now to FIGS. 11A-C, in FIG. 11A, E² distribution for radiationat 730 nm wavelength in front layers twenty through forty-one of a theNGDD-mirror structure of FIG. 4 is schematically, graphically depicted.The width of the layers is, in FIGS. 11A-C and in similar graphs herein,in general proportion to their physical thickness. It can be seen thatE² peaks are much higher than illustrated for the quarter-wave mirror ofFIG. 10, and that this is due to resonant amplification of E² in the“conjugate cavity” formed about the junction of layers 36 and 37. Notethat the second E² peak 72 (at this junction) is higher than the firstE² peak 71 and represents the highest intensity at this wavelength inthe structure. For layers deeper than this junction, the E² peaks areprogressively attenuated, there being no observable resonant effect (atthis wavelength) from the conjugate cavity centered about the junctionof layers 36 and 37. E² at layer number 22 of this structure is aboutequal to that at layer number 30 of the quarter-wave mirror of FIG. 10.The deeper penetration of the electric field in the structure of FIG. 4is clearly caused by the resonant mechanism of the conjugate cavity.

Referring now to FIG. 11B, E² distribution for radiation at 790 nmwavelength in front layers twenty through forty-one of a the NGDD-mirrorstructure of FIG. 4 is schematically, graphically depicted. It can beseen that resonance around the junction of layers 36 and 37 is muchstronger than at the 730 nm wavelength as indicated by E² peak 82. Itcan also be seen that the attenuation of subsequent peaks is interruptedby the onset of resonance in the conjugate cavity formed around thejunction of layers 31 and 32. This is evidenced by the nearly-equal peakintensity of E² peaks 83 and 84.

Referring now to FIG. 11C, E² distribution for radiation at 870 nmwavelength in front layers twenty through forty-one of a the NGDD-mirrorstructure of FIG. 4 is schematically, graphically depicted. Here it canbe seen that resonance at the conjugate cavity formed around thejunction of layers 31 and 32 becomes stronger than at 790 nm asevidenced by peak 92. At the junction of layers 36 and 37 there is stillsome trace of resonance, although it is not clear to which conjugatecavity it can be attributed.

In the foregoing discussion, an NGDD-mirror structure in accordance withthe present invention has been described in terms of a uniquedouble-resonant layer structure which reproduces the NGDD performance ofthe best prior-art NGDD-mirror of Szipocs et al. without any adverseeffects resulting from the deliberate inclusion of resonant mechanismsin the structure, which mechanisms Szipocs et al. teach should beeliminated or avoided. Further, the inventive NGDD structure greatlyexceeds the reflection performance which is obtained in that prior-artstructure. Set forth below with reference to FIG. 12 and FIG. 13 is adescription of how orderly, multiple-resonant NGDD-mirror structures inaccordance with the present invention can be used to provide a muchgreater nearly-constant NGDD than is provided by the above-discussedstructures.

FIG. 12 illustrates, in bar graph form, an inventive NGDD structurederived by computer optimizing (to a target NGDD value of −80 fs² and areflectivity of 100% using MACLEOD™ software) a triple-resonant initialstructure of the form:

(L/2 H L/2)¹⁵(H/2 L H/2)⁴(L/2 H L/2)³(H/2 L H/2)²  (3)

where L and H are as described above for structure 1, with the leftmostlayer being on the substrate, i.e., odd-numbered layers arelow-refractive-index layers. The computed GDD as a function ofwavelength for the structure of FIG. 12 is schematically depicted inFIG. 13. The GDD varies less than ±10% about a nominal value of about−78 fs² between about 740 and 840 nm. Reflectivity over this range isgreater than about 99.99%.

Those skilled in the art will recognize from FIG. 12 that as a result ofoptimization, the initial layer group (L/2 H L/2)³ of initial structure(3) has been altered in form such that, in effect, there is an L/2equivalent layer of the group “missing”. The original junction of thecavity now appears to be defined by only one (high refractive index)eighth-wave layer 41. It should be noted, however that this is stillsufficient to form the center of a virtual-cavity or conjugate-cavity.An empirical description of a mechanism for this is described in detailfurther below.

In FIG. 14 is schematically depicted, in bar graph form, a structurewhich was computer optimized using the computed reflection phase as afunction of wavelength of the structure of FIG. 12 as a target, andusing an initial structure

(L/2 H L/2)¹⁵(H/2 L H/2)⁴(L/2 H L/2)²(H/2 L H/2)²  (4)

where H and L are as specified above for the structure of FIG. 4 andodd-numbered layers are low-refractive-index layers. The variable-metricoptimization method of TFCalc™ was used for the optimization, withphase-shift only as an optimization target.

It can be seen that, following this optimization, the general form ofinitial structure is still clearly recognizable. The computed GDD as afunction of wavelength for the structure of FIG. 14 is essentiallyexactly as schematically depicted in FIG. 13, as might be expected. Fromthe above-presented description and consideration of the structure ofFIG. 14, resonance peaks for E2 would be expected near the junction oflayers 43 and 44; 38 and 39; and 29 and 30.

This expectation is verified by FIGS. 15A-C which schematicallyillustrate electric field distribution for radiation at about 772, 825,and 845 nm wavelength, respectively, in front layers twenty-four throughforty-eight of a the NGDD-mirror structure of FIG. 14. In FIG. 15Aresonance peaks 122 and 124 occur about the junction of layers 43 and44, and 38 and 39 respectively. In FIG. 15B resonance peaks 126 and 127occur about the junction of layers 38 and 39, and 29 and 30respectively. In FIG. 15C resonance peaks 129 and 130 occur about thejunction of layers 43 and 44, and 29 and 30 respectively.

It should be noted here that the electric field sampling wavelengths ofFIGS. 15A-C are chosen to demonstrate that resonance does in fact occurwhere it might be expected. Those skilled in the art who chose toinvestigate a similar structure in more detail at any other wavelengthswithin the effective constant GDD bandwidth will find that resonancealways occurs about at least one of the three junctions, and only aboutone or more of these three junctions. Here, it should be kept in mindthat a conjugate-cavity characteristic is that the resonance peak occursat one side or another of the actual interface between the dissimilareighth-wave (near eighth-wave after optimization) layers.

A remarkable feature of the inventive layer structures depicted in FIGS.4, 12 and 14, considering teachings to the contrary of the prior art, isthat most of the layers have an optical thickness close to a QWOT at thenominal (frequency) center of the reflection band produced thereby. Byway of example, in FIG. 4, 34 of 41 layers (more than 80%) are within10% of a 790 nm QWOT, and in FIG. 14, 40 of 48 layers are within ±10% ofa 790 nm QWOT. In what can be defined as the mirror sections of thesestructures all layers but the first are within ±10% of a QWOT. Inpractice this first layer may be omitted without any significant effecton NGDD performance.

Further, it will be noted from FIGS. 4, 12, and 14 and correspondingelectric field penetration data of FIGS. 15A-C and the like, that manylayers on the substrate side of a structure (the mirror portion) are notneeded to create the desired NGDD effect but are merely included toboost reflectivity. Accordingly, those skilled in the art who choose toexperiment with the structures and principles of the present inventiontaught herein will find that, within limits, the front, NGDD-formingportion of a structure may retain a closely similar structure if QWOTlayers are added to or subtracted from the mirror portion to raiseoverall reflectivity. Further, it is possible to select a commonthickness of those essentially QWOT layers near the substrate which willalign the peak of a reflection band with the center of a useful range ofNGDD. It is even possible to replace a group of one-QWOT layers with alayer of a highly-reflective metal and still achieve at least about thesame NGDD performance and reflectivity of the prior-art, Szipocs et al.structure.

Such a structure is illustrated, in bar-graph form, in FIG. 16 whereinodd-numbered layers are high-refractive-index layers. The structure isdepicted as deposited on a silver (Ag) layer (not numbered). Thisstructure was derived by eliminating layers 2-19 of the alreadyoptimized structure of FIG. 4, setting the remaining layers on an opaquelayer of silver and re-optimizing to the reflection-phase as a functionof wavelength for the structure of FIG. 4, which, of course, providesessentially the same GDD performance. In this regard, the similarity ofthe thickness relationship of layers 13-23 of the structure tocorresponding layers 31-41 of FIG. 4 and corresponding layers 38-48should be noted. Reflection as a function of wavelength for thestructure of FIG. 16 is schematically depicted in FIG. 17 and can becompared with the reflectivity obtained by Szipocs (FIG. 2) with almosttwice as many layers but with essentially identical GDD performance.Similar structures may be derived for layers of other highly-reflectivemetals, such as aluminum (Al), magnesium (Mg) or gold (Au). Structuresmay be deposited on a polished substrate of a highly-reflective metalrather than a on layer of the highly-reflective metal on some othersubstrate.

In initial designs for NGDD-mirrors, preference has been given to theindex-conjugate effect, resulting from the juxtaposition of eighth-wavelayers by juxtaposition of (H/2 L H/2)^(n) and (H/2 L H/2)^(m) groups(where n and m can be equal or different integers), for forcing theresonance in the inventive structures which provides the NGDD with aminimum amount of layers. As observed with reference to FIG. 12 however,resonance can be created when a single layer about one-eighth wavelengthor less in optical thickness is located between two significantlythicker layers. This is explained below with reference to FIG. 18. Here,a simple 11-layer resonant structure is shown that was created bybeginning with a structure (H L)⁵ H at 790 nm, where H and L have thesame values as in other above-discussed structures. The beginningstructure was optimized to a single target of 4.25% reflectance at 790nm. Layers five, six and seven were allowed to be variable. Layer numbersix was constrained to be variable only in a thickness range from 0.3 to1.0 QWOTS, to avoid optimizing to a Fabry-Perot cavity by eliminatingthis layer and consolidating the bounding layers. The E² distribution inthe structure, at 790 nm, is shown in FIG. 19.

The structure of FIG. 18 was designed to induce resonance at the center(QWOT) wavelength (790 nm) of the layers in the structure. From asimilar structure resonance can be induced by the simple mechanism ofreducing the thickness of only one of the layers. This concept isillustrated in FIG. 20 which graphically depicts (curves 20A-E)reflectivity as a function of wavelength for a symmetrical eleven-layerstructure (HL)²HfLH(LH)², with QWOT at 790 nm wavelength, for values off of 0.1, 0.2, 0.3, 0.4 and 0.5 respectively. Curve 20F depicts thereflectivity as a function of wavelength for the “initial”,eleven-layer, all-quarter-wave mirror, i.e., where f=1.0. Curves 20A-Ehave reflection minima M1-5, respectively, which lie within thereflection bandwidth of the initial structure. Those familiar with theart will recognize that when f=0.0 the structure reduces to a nine-layerFabry-Perot structure (HL)² 2H(LH)² which would have minimum reflectionand maximum transmission at 790 nm. Those familiar with the art shouldalso recognize that all of the minima M1-5 are the result of resonanceeffects similar to those which provide minimum reflection and maximumtransmission in the Fabry-Perot type structure. Those familiar with theart should further recognize that in asymmetrical layer arrangements(with more layers on one side of a virtual cavity than the other)reflection minima will occur at higher values. As these minima arecreated by the resonance effect it can be seen how the thickness of thesub-QWOT layers in the inventive structures is used, in optimizing, to“tailor” selective resonant-trapping of certain wavelengths as a meansof providing the required negative GDD.

It has been found that in designing above-described NGDD-mirrorstructures in accordance with the present invention, at least an initialcomputer optimization can be made by first manually providing a startingstructure by entering an equal-optical-thickness (quarter-wave) stack,of the type (H L)^(n)H, for a desired operating wavelength range andreflectivity, then manually reducing the thickness of at least the firstand one or more other layers, for example, the fifth and eleventh, toprovide an initial placement of resonant cavities in the stack. Thesewill initially manipulate the phase-dispersion into a general formrequired to provide high NGDD, however, requiring further refinement toprovide nearly-constant negative GDD. From this initial insertion ofresonant effective, virtual, or conjugate-cavities, computer (automatic)optimization will proceed much more rapidly and use primarily outermostlayers to achieve, as closely as is possible, the desired NGDD (andreflectivity). This then leaves the remaining layers at near-equaloptical thickness for example within about 10% of some nominal averagevalue, for providing as high as possible a reflectivity from the numberof layers and their refractive index.

Having started optimizing in this way, and depending on the optimizationalgorithm and target settings, the resulting optimized stack should befairly orderly and have a large percentage of adjacent layers about thesame optical thickness. In this regard, it may be found useful to allow,at least initially, only the outermost 15 to 20 layers to be optimized.If the optimized structure resembles one of the orderly forms discussedabove or some other recognizable resonant structure, such a structurecan be used for as a starting point (starting-structure) for a secondoptimization, which, in most instances will provide an even more orderlyoptimized structure. Also, once a satisfactory GDD performance isobtained, it is then possible to create orderly structures, as notedabove, by optimizing to the corresponding reflection-phase-shiftversus-wavelength (phase-dispersion) curve, rather than GDD. In thisway, depending on processor-speed of whatever computer is used,structures can be evaluated in minutes rather than hours, or even daysof computing time, even if as many as twenty or thirty targetphase-values are used.

Up to this point in this description of the inventive NGDD-mirrorstructures, the structures described are those which provide anearly-constant GDD at a desired reflectivity over a wavelength range orbandwidth within (less than) what might be referred to as the “normal”or characteristic reflection bandwidth associated with the high and lowrefractive index values of layers of the structure. Simply defined, thisnormal reflection bandwidth is the bandwidth of an all-quarter-wavereflector stack. The level at which the bandwidth is measured can bearbitrarily defined, for example: at 99.0% reflectivity or greater; at99.9% reflectivity or greater, or at whatever level is appropriate. Asnoted above, an all-quarter-wave stack also provides the highest peakreflectivity available with a given number of layers having thoseindices.

Clearly, there will be-applications in which NGDD-mirrors operable overa bandwidth greater than the “normal mirror bandwidth” will be ofadvantage. NGDD-mirror structures in accordance with the presentinvention can be designed to operate at such an extended bandwidth. Oneexample of such a structure is described below, beginning with referenceto FIG. 21, FIG. 22, and FIG. 23.

FIG. 21 is a bar chart schematically illustrating optical thickness oflayers of a forty-eight layer structure in accordance with the presentinvention. Odd and even-numbered layers have respectively the 2.31 highrefractive index and 1.45 low refractive index of other examplesdescribed above. In FIG. 21, it should be noted that QWOTS are at 730 nmrather than the 790 nm of other above-described examples.

FIG. 22 is a graph illustrating the computed reflection as a function ofwavelength for the NGDD-mirror structure of FIG. 21. FIG. 23 is a graphschematically illustrating the computed reflection-GDD as a function ofwavelength for the NGDD-mirror structure of FIG. 21. It can be seen thatthe reflection of the structure of FIG. 22 is greater than 99.9 percentbetween about 720 nm and 1000 nm, i.e., a 280 nm bandwidth at thislevel. From FIG. 23 it can be seen that reflection-GDD is relativelynearly-constant (within about ±10%) about a reflection-GDD of −50 fs²from about 770 nm to 1000 nm, i.e. an NGDD bandwidth of about 230 nm.

Referring now to FIG. 24, the computed reflectivity (curve 21A) andphase-shift on reflection (curve 21B) of a prior art quarter-wave stackof 27 layers ((H L)¹³H @850 nm) of index 2.31 and 1.45 is depicted.Comparing this with the reflectivity illustrated in FIG. 22 for thestructure of FIG. 21, it can be seen that, while the peak reflectivityis comparable, the 99.9% -bandwidth of the quarter-wave stack extendsonly from about 760 nm to 970 nm, i.e., a bandwidth of 210 nm. Comparingthe reflectivity of the structure of Szipocs, depicted in FIG. 2, it canbe seen that the 99.9% bandwidth of Szipocs' structure extends from 740nm to 900 nm, i.e., a bandwidth of about 160 nm, or about 40 nm lessthan a comparable “normal” mirror. The Szipocs structure (FIG. 3) has a−45 fs² NGDD bandwidth of about 140 nm. So it can be seen that thestructure of FIG. 21 provides a reflection bandwidth and a NGDDbandwidth which each exceed the comparable reflection bandwidth of anormal mirror, and far exceed that of the prior-art Szipocs structure.

The structure of FIG. 21 can be analyzed, as discussed above for otherexamples of the inventive NGDD-mirrors, by considering it as having afront portion primarily taking care of providing a desired reflectivityand reflection bandwidth, and a rear (mirror or reflector) portionprimarily taking care of providing the high phase-dispersion (from aminimum of π to one or more cycles of 2 π) and phase-dispersion shapingnecessary to provide a desired constant NGDD over a desired bandwidth.Optionally, a few intermediate layers may be considered as “interface”layers. Surprisingly, in the mirror portion of the structure of FIG. 2,more than eighty-percent of the layers have an optical thickness ofabout 1.15 QWOT at 730 nm. This is equivalent to an optical thickness ofabout one QWOT at about 840 nm, which is within the operative wavelengthrange near the frequency-center thereof. As in all other above-describedexamples of the inventive NGDD-mirrors, structure 21 relies on resonantmechanisms or sub-structures in the front portion of the structure toprovide the desired NGDD property.

Beginning with an analysis of the front portion of structure 21, FIGS.25A and 25B show electric field (E²) distribution in layers 26-48 of thestructure of FIG. 21 at wavelengths of respectively 950 nm and 1000 nm.Referring again to FIG. 21, there are actually five resonant (conjugateor virtual) cavities in this group of layers. Three of these cavitiesare clearly evident about layers 45, 41 and 37. Two others are slightlyless evident around layers 33, and 31. The resonant effect of the lattertwo, however, is clearly evident from the electric field amplificationcaused thereby, as depicted in FIGS. 25A and 25B respectively.

Referring now to FIG. 26, the computed reflection (curve 23A) andphase-shift on reflection (curves 23B) as a function of wavelength forthe NGDD-mirror structure of FIG. 21 are graphically depicted. Theeffect of the multiple resonant cavities of front portion of thestructure of FIG. 21 are clearly evident in that the phase-shift onreflection varies by almost three cycles of 2 π (three cycles of 360°)within the above-mentioned reflection bandwidth, with, as would beexpected, constantly increasing slope. Two slight reflection “dips” incurve 23A at about 770 nm and 800 nm should be noted. The cause of thesedips is discussed further hereinbelow.

Referring now to FIG. 27, the computed reflection (curve 24A) andphase-shift on reflection (curves 24B) as a function of wavelength forthe front portion (selected as layers 28-48) of the NGDD-mirrorstructure of FIG. 21 (as it would appear if deposited directly on aglass substrate) are graphically depicted. This front portion may bereferred to as the phase-retarder portion or NGDD-portion. Two aspectsof FIG. 27 are notable. First, reflection curve 24A transitions smoothlyand slowly from a relatively low reflection value of less than 20% atabout 1000 nm, only making substantial contribution (>90%) toreflectivity at wavelengths shorter than about 840 nm. This portionclearly provides the required almost-three-cycles of 2 π phase-shift. Itshould be noted, however, that the phase-shift-slope increases up towavelengths of about 900 nm and then begins to decrease slightly.

Referring next to FIG. 28, the computed reflection (curve 25A) andphase-shift on reflection (curves 25B) as a function of wavelength forthe rear portion (selected as layers 1-27) of the NGDD-mirror structureof FIG. 21, as it would appear if layers 28-48 were not depositedthereon. The following should be noted.

Comparing with FIG. 24, the phase-dispersion behaves in the same way asa “normal” quarter-wave stack but over a significantly wider bandwidth.From an abrupt change in slope at a short-wavelength band-extremity,phase-shift slope first gently decreases to about 0°/360° at 850 nm andthen gently increases toward another abrupt change in slope beyond 1000nm. It is effectively the combination of curves 24B of FIG. 27 andcurves 25B of FIG. 28 which provide curves 23B of FIG. 26. The constantNGDD bandwidth limit results from the abrupt phase changes of this rear,mirror-portion of the structure of FIG. 18. In the optimization, thephase-form of the mirror portion is taken into account in structuringthe phase-retarder portion.

Continuing with reference to FIG. 28, the reflection is generally highand at the required level across this bandwidth with the exception of adip in reflection between about 750 and 820 nm. The dip in reflection inthis region is, of course, mostly remedied by the short-wavelengthcontribution of phase-retarder layers 28-48, although, as noted above,traces are still observable in reflection curve 23A of FIG. 26.

This dip is an artifact of the bandwidth-broadening mechanism (andphase-broadening mechanism) for this mirror which, in this instance, isa multiple-resonant mechanism traceable to increased-thickness layers 3,7, and 13 of this mirror portion of the structure of FIG. 21. Layers 37, and 13 have an optical thickness between about three-eighths andseven-eighths wavelength at some wavelength within the operativereflection band of the structure. This provides the above-notedreflection minima at 770 and 800 nm. Were one to examine theelectric-field intensity distribution around layers 3, 7, and 13, onewould find resonant amplification of the field, although, of course, theactual magnitude of the amplified field intensity would be about three,four, or more orders of magnitude less than in the resonant regions ofthe phase-retarder portion of the structure. When calculated from adirection opposite from the direction of intended use, i.e., as would beseen from the rear, reflection phase-change undergoes two cycles of 2 πin the region between about 750 to 820 nm characteristic of the resonantstructure.

It is somewhat surprising and convenient that this rearward location ofthe thick layers can enable use of a resonant mechanism for reflectionbandwidth broadening and at the same time smoothly extend the nearlyflat phase-change region as seen from the direction of use withoutadversely affecting the NGDD performance of the mirror. In a simpletest, it was found that reducing the increased thickness layers to aboutthe same thickness as surrounding layers had the effect only of reducingthe GDD and reflection bandwidth, the GDD in the reduced bandwidthremained essentially the same.

It is believed that “hiding” the band broadening substructure (forexample layers 1-14) of the reflector portion of the structure of FIG.21, (or any other broad-band mirror structure) “behind” a block oflayers of about equal optical thickness (for example, layers 15-27 ofFIG. 21), or in a block of layers generally gradually graded in opticalthickness from front to back or vice-versa, but wherein any adjacent twolayers has about equal optical thickness (for example, a difference lessthan 5%), is important in ensuring that the mirror or reflector portionof the inventive structure has the desired, smoothly-varying andrelatively-flat phase-shift characteristic described above. Generally itwill be found that the thinnest and thickest layers in such a gradedblock of layers have an optical thickness respectively greater than andless than about one quarter-wavelength at respectively the shortest andlongest of the selected band of wavelengths for the NGDD-mirror.

This block or sub-structure of the broad-band mirror is of course (anyinterface layers aside) that block on which the phase-retarder structureis superposed. In an alternative definition, this block or sub-structureshould not, itself, include any resonant sub-structure for wavelengthswithin the operative wavelength range of the mirror. Behind thissub-structure, depending on broadening required, there may be one ormore spaced-apart layers having an optical thickness between about threeeights and seven-eighths wavelength at a wavelength within the operativereflection band of the inventive NGDD-mirror with one or more layerstherebetween having an optical thickness less that one quarterwavelength at a wavelength within the operative reflection band of theinventive NGDD-mirror.

It will be evident to those skilled in the art that this and other abovedescribed structures of NGDD-mirrors in accordance with the presentinvention provide considerable freedom in selecting reflectivityspecifications and NGDD specifications, while not completelyindependently, in a only a loosely-coupled way. Certainly, knowledge ofthe which essential structural features are most important incontrolling and defining particular ones of the mirror specifications isof great value in making sure that computer optimization provides asorderly as possible a structure for ease of manufacture. Having a priorknowledge of basic structural forms significantly reduces the timerequired, and the success percentage achieved, in computer optimizingsuch structures compared with methods taught in the prior art.

In this regard, having a knowledge of required characteristics of themirror portion of the mirror function provides that a broad band NGDDmirror in accordance with the present invention can be designed in twoseparate steps. A first step would be to design a broad-band mirrorportion by specifying a required phase characteristic as well as areflectivity characteristic from a first initial structure. A secondstep would be adding to the broad band mirror structure, a sufficientnumber of layers including two or more resonant substructures arrangedto provide a desired NGDD characteristic. A third step would be tooptimizing the entire structure to meet required reflectivity and NGDDtargets.

It should be noted that the inventive NGDD-mirrors, by virtue of thecomplex, multiple-resonant structure which is required to provide theexemplified magnitude and bandwidth of negative GDD, are sensitive tolayer thickness errors. They can be expected to present the similarmanufacturing challenges as other complex optically-resonant layerstructures such as high-selectivity bandpass filters with multiple,coupled, resonant-cavities. In these inventive structures, however, bymaintaining those layers which have limited ability to influence GDD orextend reflection or GDD bandwidth at near-equal thickness, andcertainly by making sure that adjacent ones of these layers are as closein optical thickness as possible, this sensitivity can be significantlyreduced compared with the prior-art Szipocs et al. structures. This isparticularly true, for example, of structures of the type depicted inFIGS. 4, 12, and 14, wherein essentially all layers in a mirror portionof the structure have about equal optical thickness.

In examples of NGDD-mirror structures discussed above, initialstructures have been optimized, which are formed from only twomaterials, one having a relatively high refractive index and the other arelatively low refractive index. Relatively high and low, here, beingtypical for essentially (low loss) materials transparent in theexemplified wavelength ranges. Optimization has been made against fixedNGDD and reflection parameters as might be specified for a particularlaser device. One result of this, which is found in all suchoptimizations, is that in the nearly-constant NGDD band, there is avariation, or ripple, indicating third or higher order contributions ofthe phase-dispersion of the structures. In many applications, thisrippling or lack of smoothness may be found adequate or tolerable. Byintroducing additional optimization variables, however, it may bepossible to reduce the rippling, i.e., provide a more nearly-constantNGDD, albeit, in most instances, by adding structural complexity.

A simplest smoothing possibility, is to allow the GDD and reflectionproperties (magnitude and bandwidth) to be variable in a structure witha favored pair of materials and a fixed maximum number of layers(maximum, here, recognizing that even a purely thickness-varyingoptimization algorithm can reduce the number of layers by virtue of azero-thickness option). This is simply trading greater NGDD-smoothnessfor lesser conformance with one or more other desired parameters andadds no complexity to the structure.

More complex, is to use the refractive index of layers, particularly inthe phase-retarder portion of the structure, as an optimizationvariable. This can be done in a real or “simulated” way. Examples ofthis are set forth below.

In a first example, “real” index variation, the refractive index(simply) can be made an optimization variable. It should be noted here,however, that if the refractive index is significantly varied it mayalso be necessary to make additions to the number of layers in aparticular localized group of layers in a structure. For example, in astructural group such as (H/2 L H/2)² where H has the refractive indexvalue 2.31 and L has the refractive index value 1.45, it may benecessary to change the sub-structure of the group itself to (H/2 LH/2)³ if the refractive index value of H is varied to 1.9. Those skilledin the art will be aware that there are optimization algorithms whichallow layers to be automatically be added to a structure to reach aparticular target. Alternatively, the group repetition number may befixed and a third group (another virtual cavity) added. This, of courseinvolves adding complexity at least by virtue of the number of layers.Adding layers in the phase-retarder section, however, may in itself beadvantageous, as it provides more individual phase-components which canbe summed to provide the phase (smooth NGDD) form required.

A significant restriction in index-variation is that useful “real”materials are typically available only in a limited range of refractiveindices, with values not equally distributed throughout that range.Deposition of mixture of two materials may be used to at least partiallyovercome this restriction. This can add to manufacturing, if notstructural complexity.

Adding layers and providing index variability can be simultaneouslyaccomplished (manually), for example, by the technique ofmultilayer-group substitution for a single layer (preferably about aquarter-wave layer) in the phase-retarder group. Those skilled in theart will be aware that in one particular implementation of thetechnique, this involves (initially) substituting a combination of threelayers for the single layer. The outermost of the layers may have ahigher refractive index that the original layer, and the inner layer alower refractive index, or vice-versa. The combination has about thesame or somewhat lesser optical thickness than the single layer. Thesubstitution adds boundaries which can be effective in NGDD optimizationas well as providing a means of varying effective index of thesubstituted layer (by varying thickness of the component layers) withlayers of preferred “real” materials.

Another technique which may be used for providing a structure forsmoothing of NGDD consistent with principles of the present invention,is to optimize an initial or starting structure wherein, in thephase-retarder portion of the structure, one or more virtual cavitiesare formed, for example, by a conjugate layer-grouping of the form:

(aL bM cH bM aL)^(n)(aH bM cL bM aH)^(n)  (5)

where 2a+2b+c=2.0 and M is a QWOT of a material having a refractiveindex intermediate that of H and L. Those skilled in the art willrecognize that this technique may be extended to more complexrepetition-groups including two or more intermediate index layer-pairsarranged in a graded fashion, or, in the limit, where each side of theconjugate grouping is a single periodic-inhomogeneous layer. This wouldbe facilitated, of course, by optimization software able to optimize theindex/thickness function of the layers.

Of course, any of the above-discussed variable options may be used aloneor in combination for optimizing any particular parameter of anNGDD-mirror in accordance with the present invention. From the forgoingdescription, those skilled in the art may contemplate several suchcombinations of optimization or structural variables without departingfrom the spirit and scope of the present invention.

In all above-discussed examples of NGDD-mirrors in accordance with thepresent invention, structures have been designed with hypotheticalmaterials which have neither dispersion nor loss. This is done in orderto make a direct comparison of structural principles alone withanalyzable prior-art approaches and between the examples themselves. Infact, were any of the above described structures to be re-optimized foran equivalent “real” material with a refractive index which variesacross the operative bandwidth through dispersion, for example niobiumoxide (Nb₂O₅) for index 2.31, and silicon dioxide (SiO₂) for index 1.45,there would be very little difference in the exemplary structures. Whatdifference would exist would be a slight departure, for example about1%, from exact equality of those layers which are exemplified as exactlyequal in optical thickness, and some comparable difference in thicknessof corresponding layers in the phase-retarder portions andbandwidth/phase-broadening portions.

Regarding zero loss, using a low-loss deposition technique, particularlyion-beam sputtering, can provide losses-per-layer sufficiently low thatthey have no distinguishable effect on form or function the inventivestructures. By way of example, using ion-beam-sputter-deposited layersof certain high- refractive-index materials such as tantalum oxide(Ta₂O₅) or hafnium oxide (HfO₂), and a low refractive index materialsuch as SiO₂, a total optical loss less than about 5×10⁻⁵ (0.005%) maybe obtained for a stack of about forty, or even more, alternatinglayers.

In all above-discussed examples of NGDD-mirrors in accordance with thepresent invention, structures have been optimized for use with light(radiation) incident normally thereon, and the optical thickness oflayers is the optical thickness at that (normal) angle of incidence.Again, this is done in order to make a direct comparison of structuralprinciples alone with analyzable prior-art approaches and between theexamples themselves. Principles of the present invention are equallyapplicable, however, if structures are designed for use at non-normalincidence. Corresponding structures may be similarly defined in opticalthickness terms given that optical thickness is the effective opticalthickness for a particular polarization orientation, at theangle-of-incidence of intended use.

In certain instances, there could be some advantage to designingstructures for use at a relatively high incidence angle, for example,about 45°, and for light polarized perpendicular to the plane ofincidence (s-polarized). This has been done with prior-art NGDD devices,such as the Gires-Tournois interferometer, to extend the bandwidth ofthe device.

An advantage of designing for s-polarized light is that for a given pairof materials (a high and a low refractive index material), the effectiveindex-ratio for the s-polarized light increases with increasing angle ofincidence. This means that the above-discussed “normal” reflectionbandwidth for these materials increases. Those familiar with the artwill recognize that the individual effective-indices actually decrease.This can be advantage in exploiting some of the above-discussedsmoothing options, inasmuch as the angle of incidence can provideeffective-index variation with preferred “real” materials.

Further, a relatively low-index-ratio material-pair, for example HfO₂and SiO₂, at some practical angle of incidence, will have the sameindex-ratio as, say, Nb₂O₅ and SiO₂ at normal incidence. At thisincidence-angle Nb₂O₅ would be effective as a material withsignificantly higher refractive index at normal incidence. Theprinciples of non-normal incidence layer system design are well-known tothose skilled in the art and a further more detailed description ofapplying the inventive principles is not necessary. Accordingly, nospecific examples of the inventive NGDD-mirrors at non-normal incidenceare presented herein.

Throughout the above-presented description, examples of NGDD mirrorstructures in accordance with the present invention have been presentedwith reference to wavelength-dependent functions for parameters such asreflectivity, NGDD and reflective phase-shift, as such are common in theart to which the present invention generally pertains. Those skilled inthe art will recognize that the principles of the present invention areequally applicable if a wavelength related parameter such as frequency,energy, wavenumber or relative-wavenumber is selected for specifyingthose parameters.

The present invention has been described and depicted as a preferred andother embodiments. The invention is not limited, however, to thoseembodiments described and depicted. Rather, the invention is defined bythe claims appended hereto.

What is claimed is:
 1. A multilayer mirror for providing greater than aselected high reflectivity value and substantially-constant negativegroup-delay-dispersion over a selected band of wavelengths, comprising:a substrate; first and second pluralities of layers disposed on saidsubstrate, with said second plurality of layers superposed on said firstplurality of layers; said first plurality of layers functioningprimarily to provide said high reflectivity; and said second pluralityof layers arranged such that selective resonant trapping of wavelengthswithin the band of wavelengths occurs in at least two longitudinallyspaced-apart cavity groups of one or more layers within said secondplurality of layers in such a way that said second plurality of layersfunctions primarily to provide a high reflection phase-dispersion forthe mirror within the selected band of wavelengths and, cooperative withsaid first plurality of layers, to provide that said reflectionphase-dispersion constantly increases over the selected band ofwavelengths from about the shortest wavelength thereof to about thelongest wavelength thereof, in a way which provides thesubstantially-constant negative group-delay-dispersion across theselected band of wavelengths.
 2. The mirror of claim 1, wherein layersin said first and second pluralities are layers of materials transparentto said selected band of wavelengths, and adjacent layers have adifferent refractive index.
 3. The mirror of claim 2, wherein said firstplurality of layers further includes a metal layer, said metal layerbeing closest to said substrate.
 4. The mirror of claim 2, wherein saidsubstrate is a metal substrate having a polished surface on which saidfirst and second pluralities of layers are deposited.
 5. The mirror ofclaim 1, wherein said first plurality of layers is essentially a stackof alternating high and low refractive index dielectric materials, eachthereof having an optical thickness of about one-quarter wavelength atwavelength within the selected range of wavelengths.
 6. The mirror ofclaim 1, wherein each of said cavity-groups includes at least one layerhaving an optical thickness less than about one-quarter wavelength at awavelength within the selected band of wavelengths.
 7. The mirror ofclaim 1 wherein at least one of said cavity groups has an opticalthickness of about one-half wavelength at a wavelength within theselected band of wavelengths.
 8. A multilayer mirror for providinggreater than a selected high reflectivity value and asubstantially-constant negative group-delay-dispersion over a selectedband of wavelengths, comprising: a substrate; first and secondpluralities of layers disposed on said substrate, with said secondplurality of layers superposed on said first plurality of layers; saidfirst plurality of layers functioning primarily to provide said highreflectivity and being essentially a quarter-wave stack of layers havingabout equal optical thickness, said optical thickness being aboutone-quarter wavelength at a wavelength within the selected band ofwavelengths; and said second plurality of layers arranged such thatselective resonant trapping of wavelengths within the band ofwavelengths occurs in at least two longitudinally spaced-apart cavitygroups of one or more layers within said second plurality of layers insuch a way that said second plurality of layers functions primarily toprovide a high reflection phase-dispersion for the mirror within theselected band of wavelengths and, cooperative with said first pluralityof layers, to provide that said reflection phase-dispersion constantlyincreases over the selected band of wavelengths from about the shortestwavelength thereof to about the longest wavelength thereof, in a waywhich provides the substantially-constant negativegroup-delay-dispersion across the band of wavelengths.
 9. A multilayermirror for providing greater than a selected high reflectivity value anda substantially-constant negative group-delay-dispersion over a selectedband of wavelengths, comprising: a substrate; first and secondpluralities of layers disposed on said substrate, with said secondplurality of layers superposed on said first plurality of layers; saidfirst plurality of layers being alternating layers of a first materialhaving a high refractive index and a second material having a lowrefractive index, said layers arranged to form a mirror having areflection band at least as wide as the selected band of wavelengths andgreater than a correspondingly-defined reflection band of a quarter-wavestack of said first and second materials; and said second plurality oflayers arranged such that selective resonant trapping of wavelengthswithin the band of wavelengths occurs in at least two longitudinallyspaced-apart cavity groups of one or more layers within said secondplurality of layers, each of said cavity groups including at least onelayer having an optical thickness less than one quarter-wavelengthoptical thickness at a wavelength within the selected band ofwavelengths, said selective resonant trapping occurring in such a waythat said second plurality of layers functions primarily to provide ahigh reflection phase-dispersion for the mirror within the selected bandof wavelengths and, cooperative with said first plurality of layers, toprovide that said reflection phase-dispersion constantly increases overthe selected band of wavelengths from about the shortest wavelengththereof to about the longest wavelength thereof, in a way which providessubstantially-constant negative group-delay-dispersion across the bandof wavelengths.
 10. The mirror of claim 9, wherein said first pluralityof layers includes first and second portions, said second portionfurthest from said substrate and characterized in that all adjacentlayer pairs therein have about the same optical thickness and whereinsaid first portion includes two or more spaced-apart layers having anoptical thickness of between about three-eighths and seven-eighths of awavelength at a wavelength within said selected band of wavelengths. 11.The mirror of claim 10, wherein all layers in said first portion of saidfirst plurality of layers each have an optical thickness of aboutone-quarter wavelength at a wavelength within the selected band ofwavelengths.
 12. The mirror of claim 11, wherein said two or morespaced-apart layers having an optical thickness of between aboutthree-eighths and seven-eighths of a wavelength at a wavelength withinthe selected band of wavelengths are spaced apart by layers having anoptical thickness of about one-quarter wavelength at a wavelength withinthe selected band of wavelengths.
 13. The mirror of claim 11, whereinbetween said two or more spaced-apart layers having an optical thicknessof between about three-eighths and seven-eighths of a wavelength at awavelength within the selected band of wavelengths are layers having anoptical thickness less than about one-quarter wavelength at a wavelengthwithin the selected band of wavelengths.
 14. The mirror of claim 10,wherein adjacent ones of said second plurality of layers have adifferent refractive index and any one layer is a layer of a materialhaving one of a high refractive index, a low refractive index, and arefractive index intermediate said high and low refractive indices. 15.A multilayer mirror for providing greater than a selected highreflectivity value and a substantially-constant negativegroup-delay-dispersion over a selected band of wavelengths in anintended direction of light incidence thereon, comprising: a substrate;first and second pluralities of layers disposed on said substrate, withsaid second plurality of layers superposed on said first plurality oflayers; said first plurality of layers functioning primarily to providesaid high reflectivity value and including alternating layers of amaterial having a high refractive index and a material having a lowrefractive index, said layers arranged to form a mirror having areflection band at least as wide as the selected band of wavelengths,and wherein said first plurality of layers alone, seen in the intendedlight-incidence direction of the mirror would have a phase-dispersiondefined as reflection phase-shift as a function of wavelength varying byno greater than π/2 over said selected band of wavelengths, the form ofsaid reflection-phase-shift function being smoothly and monotonicallyvarying, said high and low index values of said first plurality oflayers being such that a quarter-wave-stack of such layers having acenter wavelength within said selected band of wavelengths would have abandwidth about equal to or less than the selected band of wavelengths;and said second plurality of layers arranged such that selectiveresonant trapping of wavelengths within the band of wavelengths occursin at least two longitudinally spaced-apart cavity groups of one or morelayers within said second plurality of layers, each of said cavitygroups including at least one layer having an optical thickness lessthan one quarter-wavelength optical thickness at a wavelength within theselected band of wavelengths, said selective resonant trapping occurringin such a way that said second plurality of layers functions primarilyto provide a high reflection phase-dispersion for the mirror within theselected band of wavelengths and, cooperative with said first pluralityof layers, to provide that said reflection phase-dispersion constantlyincreases over the selected band of wavelengths from about the shortestwavelength thereof to about the longest wavelength thereof, in a waywhich provides substantially-constant negative group-delay-dispersionacross the selected band of wavelengths.
 16. The mirror of claim 15,wherein said high phase-dispersion created by said second plurality ofplayers is greater than about π over the selected band of wavelengths.17. The mirror of claim 15, wherein said second plurality of wavelengthsincludes alternating layers of a high refractive index material and alow refractive index material.
 18. The mirror of claim 17, wherein saidhigh and low refractive index materials of said second plurality oflayers are respectively the same as said high and low refractive indexmaterials of said first plurality of layers.